The Hardy–Littlewood Maximal Operator

2.1 Maximal Functions

2.1.1 The Hardy–Littlewood Maximal Operator

Definition 2.1.1.The function M(f)(x) =sup

δ>0

Avg

B(x,δ)

|f|=sup

δ>0

1 v_nδⁿ

Z

|y|<δ|f(x−y)|dy is called thecentered Hardy–Littlewood maximal functionof f.

Obviously we haveM(f) =M(|f|)≥0; thus the maximal function is a positive operator. Information concerning cancellation of the function f is lost by passing toM(f). We show later that M(f)pointwise controls f (i.e.,M(f)≥ |f| almost everywhere). Note thatMmapsL^∞to itself, that is, we have

M(f)

L^∞≤ f

L^∞.

Let us compute the Hardy–Littlewood maximal function of a specific function.

Example 2.1.2.OnR, let f be the characteristic function of the interval[a,b]. For x∈(a,b), clearlyM(f) =1. Forx≥b, a simple calculation shows that the largest average of f over all intervals(x−δ,x+δ)is obtained whenδ=x−a. Similarly, whenx≤a, the largest average is obtained whenδ=b−x. Therefore,

M(f)(x) =







(b−a)/2|x−b| whenx≤a,

1 whenx∈(a,b),

(b−a)/2|x−a| whenx≥b.

Observe thatM(f)has a jump atx=aandx=bequal to one-half that off. Mis a sublinear operator and never vanishes. In fact, we have that ifM(f)(x₀) = 0 for somex₀∈Rⁿ, then f =0 a.e. Moreover, if f is compactly supported, say in

|x| ≤R, then

M(f)(x)≥ f

_L1

v_n 1

(|x|+R)ⁿ, (2.1.1)

for|x| ≥R, wherev_nis the volume of the unit ball inRⁿ. Equation (2.1.1) implies thatM(f)is neverinL¹(Rⁿ)if f 6=0 a.e., a strong property that reflects a certain behavior of the maximal function. In fact, ifgis inL¹_locandM(g)is inL¹(Rⁿ), then g=0 a.e. To see this, use (2.1.1) withg_R(x) =g(x)χ_|x|≤Rto conclude thatg_R(x) =0 for almost allxin the ball of radiusR>0. Thusg=0 a.e. inRⁿ. However, it is true thatM(f)is inL^1,∞when f is inL¹.

A related analogue ofM(f)is its uncentered versionM(f), defined as the supre- mum of all averages of f over all open balls containing a given point.

Definition 2.1.3.Theuncentered Hardy–Littlewood maximal function of f, M(f)(x) = sup

δ>0

|y−x|<δ

Avg

B(y,δ)

|f|,

is defined as the supremum of the averages of|f|over all open ballsB(y,δ)that contain the pointx.

ClearlyM(f)≤M(f); in other words,Mis a larger operator thanM. However, M(f)≤2ⁿM(f)and the boundedness properties ofMare identical to those ofM. Example 2.1.4.OnR, let f be the characteristic function of the intervalI= [a,b].

Forx∈(a,b), clearlyM(f)(x) =1. Forx>b, a calculation shows that the largest average of f over all intervals(y−δ,y+δ)that containxis obtained whenδ =

1

2(x−a)andy=¹₂(x+a). Similarly, whenx<a, the largest average is obtained whenδ =¹₂(b−x)andy=¹₂(b+x). We conclude that

M(f)(x) =







(b−a)/|x−b| whenx≤a,

1 whenx∈(a,b),

(b−a)/|x−a| whenx≥b.

Observe thatMdoes not have a jump atx=aandx=band that it is comparable to the function 1+^dist_|I|^(x,I)−1

.

We are now ready to obtain some basic properties of maximal functions. We need the following simple covering lemma.

Lemma 2.1.5.Let{B₁,B₂, . . . ,B_k}be a finite collection of open balls inRⁿ. Then there exists a finite subcollection{B_j1, . . . ,B_jl}of pairwise disjoint balls such that

l r=1

∑

B_jr ≥3⁻ⁿ

k [

i=1

B_i

. (2.1.2)

Proof. Let us reindex the balls so that

|B₁| ≥ |B₂| ≥ · · · ≥ |B_k|.

Let j₁=1. Having chosen j₁,j₂, . . . ,j_i, let j_i+1be the least indexs> j_isuch that Si

m=1B_jmis disjoint fromB_s. Since we have a finite number of balls, this process will terminate, say afterlsteps. We have now selected pairwise disjoint ballsB_j1, . . . ,B_jl. If some B_m was not selected, that is, m∈ {/ j₁, . . . ,j_l}, then B_m must intersect a selected ballB_jr for some j_r<m. ThenB_mhas smaller size thanB_jr and we must haveB_m⊆3B_jr. This shows that the union of the unselected balls is contained in the union of the triples of the selected balls. Therefore, the union of all balls is contained in the union of the triples of the selected balls. Thus

k [

i=1

B_i

≤

l [

r=1

3B_jr

≤

l

∑

r=1

|3B_jr|=3ⁿ

l

∑

r=1

|B_jr|,

and the required conclusion follows.

We are now ready to prove the main theorem concerning the boundedness of the centered and uncentered maximal functionsMandM, respectively.

Theorem 2.1.6.The uncentered Hardy–Littlewood maximal function maps L¹(Rⁿ) to L^1,∞(Rⁿ) with constant at most3ⁿ and also L^p(Rⁿ)to L^p(Rⁿ)for 1<p<∞ with constant at most3^n/pp(p−1)⁻¹. The same is true for the centered maximal operatorM.

We note that operators that mapL¹toL^1,∞are said to beweak type(1,1).

Proof. SinceM(f)≥M(f), we have

{x∈Rⁿ: |M(f)(x)|>α} ⊆ {x∈Rⁿ: |M(f)(x)|>α}, and therefore it suffices to show that

|{x∈Rⁿ: |M(f)(x)|>α}| ≤3ⁿ f

L¹

α . (2.1.3)

We claim that the set

E_α={x∈Rⁿ: |M(f)(x)|>α}

is open. Indeed, forx∈E_α, there is an open ballB_xthat containsxsuch that the av- erage of|f|overB_xis strictly bigger thanα. Then the uncentered maximal function of any other point inB_xis also bigger thanα, and thusB_xis contained inEα. This proves thatEα is open.

LetKbe a compact subset ofEα. For eachx∈K there exists an open ballBx

containing the pointxsuch that Z

Bx

|f(y)|dy>α|B_x|. (2.1.4)

Observe that B_x ⊂E_α for all x. By compactness there exists a finite subcover

{B_x1, . . . ,B_xk} of K. Using Lemma 2.1.5 we find a subcollection of pairwise dis-

joint ballsB_xj

1, . . . ,B_xjl such that (2.1.2) holds. Using (2.1.4) and (2.1.2) we obtain

|K| ≤

k [

i=1

B_xi ≤3ⁿ

l i=1

∑

|B_x

ji| ≤3ⁿ α

l i=1

∑

Z

B_{x j}

i

|f(y)|dy≤3ⁿ α

Z

Eα

|f(y)|dy, since all the ballsB_xji are disjoint and contained inE_α. Taking the supremum over all compactK⊆E_αand using the inner regularity of Lebesgue measure, we deduce (2.1.3). We have now proved thatMmapsL¹→L^1,∞with constant 3ⁿ. It is a trivial fact thatM mapsL^∞→L^∞with constant 1. SinceMis well defined and finite a.e.

onL¹+L^∞, it is also onL^p(Rⁿ)for 1<p<∞. The Marcinkiewicz interpolation theorem (Theorem 1.3.2) implies thatMmapsL^p(Rⁿ)toL^p(Rⁿ)for all 1<p<∞.

Using Exercise 1.3.3, we obtain the following estimate for the operator norm ofM onL^p(Rⁿ):

M

L^p→L^p≤ p3^np

p−1. (2.1.5)

Observe that a direct application of Theorem 1.3.2 would give the slightly worse bound of 2 _p−1^{p 1}_p

3^np.

Remark 2.1.7.The previous proof gives a bound on the operator norm of M on L^p(Rⁿ)that grows exponentially with the dimension. One may wonder whether this bound could be improved to a better one that does not grow exponentially in the dimensionn, asn→∞. This is not possible; see Exercise 2.1.8.

Example 2.1.8.LetR>0. Then there are dimensional constantsc_nandc⁰_nsuch that c⁰_nRⁿ

(|x|+R)ⁿ≤M(χ_B(0,R))(x)≤ c_nRⁿ

(|x|+R)ⁿ. (2.1.6) Since these functions are not integrable overRⁿ, it follows thatM does not map L¹(Rⁿ)toL¹(Rⁿ).

Next we estimateM(M(χ_B(0,R)))(x). First we write Rⁿ

(|x|+R)ⁿ ≤χ_B(0,R)+

∞ k=0

∑

Rⁿ

(R+2^kR)ⁿχ_B(0,2k+1R)\B(0,2^kR). Using the upper estimate in (2.1.6) and the sublinearity ofM, we obtain

M

Rⁿ (| · |+R)ⁿ

(x) ≤M(χ_B(0,R))(x) +

∞

∑

k=0

1

(1+2^k)ⁿM(χ_B(0,2k+1R))(x)

≤ c_nRⁿ (|x|+R)ⁿ+

∞ k=0

∑

1 (1+2^k)ⁿ

c_n(2^k+1R)ⁿ (|x|+2^k+1R)ⁿ

≤C_nlog(e+|x|/R) (1+|x|/R)ⁿ ,

where the last estimate follows by summing separately over ksatisfying 2^k+1≤

|x|/Rand 2^k+1≥ |x|/R.Note that the presence of the logarithm does not affect the L^pboundedness of this function whenp>1.